{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "google",
   "metadata": {},
   "source": [
    "##### Copyright 2025 Google LLC."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "apache",
   "metadata": {},
   "source": [
    "Licensed under the Apache License, Version 2.0 (the \"License\");\n",
    "you may not use this file except in compliance with the License.\n",
    "You may obtain a copy of the License at\n",
    "\n",
    "    http://www.apache.org/licenses/LICENSE-2.0\n",
    "\n",
    "Unless required by applicable law or agreed to in writing, software\n",
    "distributed under the License is distributed on an \"AS IS\" BASIS,\n",
    "WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
    "See the License for the specific language governing permissions and\n",
    "limitations under the License.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "basename",
   "metadata": {},
   "source": [
    "# einav_puzzle"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "link",
   "metadata": {},
   "source": [
    "<table align=\"left\">\n",
    "<td>\n",
    "<a href=\"https://colab.research.google.com/github/google/or-tools/blob/main/examples/notebook/contrib/einav_puzzle.ipynb\"><img src=\"https://raw.githubusercontent.com/google/or-tools/main/tools/colab_32px.png\"/>Run in Google Colab</a>\n",
    "</td>\n",
    "<td>\n",
    "<a href=\"https://github.com/google/or-tools/blob/main/examples/contrib/einav_puzzle.py\"><img src=\"https://raw.githubusercontent.com/google/or-tools/main/tools/github_32px.png\"/>View source on GitHub</a>\n",
    "</td>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "doc",
   "metadata": {},
   "source": [
    "First, you must install [ortools](https://pypi.org/project/ortools/) package in this colab."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "install",
   "metadata": {},
   "outputs": [],
   "source": [
    "%pip install ortools"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "description",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "  A programming puzzle from Einav in Google CP Solver.\n",
    "\n",
    "  From\n",
    "  'A programming puzzle from Einav'\n",
    "  http://gcanyon.wordpress.com/2009/10/28/a-programming-puzzle-from-einav/\n",
    "  '''\n",
    "  My friend Einav gave me this programming puzzle to work on. Given\n",
    "  this array of positive and negative numbers:\n",
    "  33   30  -10 -6  18   7  -11 -23   6\n",
    "  ...\n",
    "  -25   4  16  30  33 -23  -4   4 -23\n",
    "\n",
    "  You can flip the sign of entire rows and columns, as many of them\n",
    "  as you like. The goal is to make all the rows and columns sum to positive\n",
    "  numbers (or zero), and then to find the solution (there are more than one)\n",
    "  that has the smallest overall sum. So for example, for this array:\n",
    "  33  30 -10\n",
    "  -16  19   9\n",
    "  -17 -12 -14\n",
    "  You could flip the sign for the bottom row to get this array:\n",
    "  33  30 -10\n",
    "  -16  19   9\n",
    "  17  12  14\n",
    "  Now all the rows and columns have positive sums, and the overall total is\n",
    "  108.\n",
    "  But you could instead flip the second and third columns, and the second\n",
    "  row, to get this array:\n",
    "  33  -30  10\n",
    "  16   19    9\n",
    "  -17   12   14\n",
    "  All the rows and columns still total positive, and the overall sum is just\n",
    "  66. So this solution is better (I don't know if it's the best)\n",
    "  A pure brute force solution would have to try over 30 billion solutions.\n",
    "  I wrote code to solve this in J. I'll post that separately.\n",
    "  '''\n",
    "\n",
    "  Compare with the following models:\n",
    "  * MiniZinc http://www.hakank.org/minizinc/einav_puzzle.mzn\n",
    "  * SICStus: http://hakank.org/sicstus/einav_puzzle.pl\n",
    "\n",
    "  Note:\n",
    "  einav_puzzle2.py is Laurent Perron version, which don't use as many\n",
    "  decision variables as this version.\n",
    "\n",
    "\n",
    "  This model was created by Hakan Kjellerstrand (hakank@gmail.com)\n",
    "  Also see my other Google CP Solver models:\n",
    "  http://www.hakank.org/google_or_tools/\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "code",
   "metadata": {},
   "outputs": [],
   "source": [
    "from ortools.constraint_solver import pywrapcp\n",
    "\n",
    "\n",
    "def main():\n",
    "\n",
    "  # Create the solver.\n",
    "  solver = pywrapcp.Solver('Einav puzzle')\n",
    "\n",
    "  #\n",
    "  # data\n",
    "  #\n",
    "\n",
    "  # small problem\n",
    "  # rows = 3;\n",
    "  # cols = 3;\n",
    "  # data = [\n",
    "  #     [ 33,  30, -10],\n",
    "  #     [-16,  19,   9],\n",
    "  #     [-17, -12, -14]\n",
    "  #     ]\n",
    "\n",
    "  # Full problem\n",
    "  rows = 27\n",
    "  cols = 9\n",
    "  data = [[33, 30, 10, -6, 18, -7, -11, 23, -6],\n",
    "          [16, -19, 9, -26, -8, -19, -8, -21, -14],\n",
    "          [17, 12, -14, 31, -30, 13, -13, 19, 16],\n",
    "          [-6, -11, 1, 17, -12, -4, -7, 14, -21],\n",
    "          [18, -31, 34, -22, 17, -19, 20, 24, 6],\n",
    "          [33, -18, 17, -15, 31, -5, 3, 27, -3],\n",
    "          [-18, -20, -18, 31, 6, 4, -2, -12, 24],\n",
    "          [27, 14, 4, -29, -3, 5, -29, 8, -12],\n",
    "          [-15, -7, -23, 23, -9, -8, 6, 8, -12],\n",
    "          [33, -23, -19, -4, -8, -7, 11, -12, 31],\n",
    "          [-20, 19, -15, -30, 11, 32, 7, 14, -5],\n",
    "          [-23, 18, -32, -2, -31, -7, 8, 24, 16],\n",
    "          [32, -4, -10, -14, -6, -1, 0, 23, 23],\n",
    "          [25, 0, -23, 22, 12, 28, -27, 15, 4],\n",
    "          [-30, -13, -16, -3, -3, -32, -3, 27, -31],\n",
    "          [22, 1, 26, 4, -2, -13, 26, 17, 14],\n",
    "          [-9, -18, 3, -20, -27, -32, -11, 27, 13],\n",
    "          [-17, 33, -7, 19, -32, 13, -31, -2, -24],\n",
    "          [-31, 27, -31, -29, 15, 2, 29, -15, 33],\n",
    "          [-18, -23, 15, 28, 0, 30, -4, 12, -32],\n",
    "          [-3, 34, 27, -25, -18, 26, 1, 34, 26],\n",
    "          [-21, -31, -10, -13, -30, -17, -12, -26, 31],\n",
    "          [23, -31, -19, 21, -17, -10, 2, -23, 23],\n",
    "          [-3, 6, 0, -3, -32, 0, -10, -25, 14],\n",
    "          [-19, 9, 14, -27, 20, 15, -5, -27, 18],\n",
    "          [11, -6, 24, 7, -17, 26, 20, -31, -25],\n",
    "          [-25, 4, -16, 30, 33, 23, -4, -4, 23]]\n",
    "\n",
    "  #\n",
    "  # variables\n",
    "  #\n",
    "  x = {}\n",
    "  for i in range(rows):\n",
    "    for j in range(cols):\n",
    "      x[i, j] = solver.IntVar(-100, 100, 'x[%i,%i]' % (i, j))\n",
    "\n",
    "  x_flat = [x[i, j] for i in range(rows) for j in range(cols)]\n",
    "\n",
    "  row_sums = [solver.IntVar(0, 300, 'row_sums(%i)' % i) for i in range(rows)]\n",
    "  col_sums = [solver.IntVar(0, 300, 'col_sums(%i)' % j) for j in range(cols)]\n",
    "\n",
    "  row_signs = [solver.IntVar([-1, 1], 'row_signs(%i)' % i) for i in range(rows)]\n",
    "  col_signs = [solver.IntVar([-1, 1], 'col_signs(%i)' % j) for j in range(cols)]\n",
    "\n",
    "  # total sum: to be minimized\n",
    "  total_sum = solver.IntVar(0, 1000, 'total_sum')\n",
    "\n",
    "  #\n",
    "  # constraints\n",
    "  #\n",
    "  for i in range(rows):\n",
    "    for j in range(cols):\n",
    "      solver.Add(x[i, j] == data[i][j] * row_signs[i] * col_signs[j])\n",
    "\n",
    "  total_sum_a = [\n",
    "      data[i][j] * row_signs[i] * col_signs[j]\n",
    "      for i in range(rows)\n",
    "      for j in range(cols)\n",
    "  ]\n",
    "  solver.Add(total_sum == solver.Sum(total_sum_a))\n",
    "\n",
    "  # row sums\n",
    "  for i in range(rows):\n",
    "    s = [row_signs[i] * col_signs[j] * data[i][j] for j in range(cols)]\n",
    "    solver.Add(row_sums[i] == solver.Sum(s))\n",
    "\n",
    "  # column sums\n",
    "  for j in range(cols):\n",
    "    s = [row_signs[i] * col_signs[j] * data[i][j] for i in range(rows)]\n",
    "    solver.Add(col_sums[j] == solver.Sum(s))\n",
    "\n",
    "  # objective\n",
    "  objective = solver.Minimize(total_sum, 1)\n",
    "\n",
    "  #\n",
    "  # search and result\n",
    "  #\n",
    "  # Note: The order of the variables makes a big difference.\n",
    "  #       If row_signs are before col_sign it is much slower.\n",
    "  db = solver.Phase(col_signs + row_signs, solver.CHOOSE_MIN_SIZE_LOWEST_MIN,\n",
    "                    solver.ASSIGN_MAX_VALUE)\n",
    "\n",
    "  solver.NewSearch(db, [objective])\n",
    "\n",
    "  num_solutions = 0\n",
    "  while solver.NextSolution():\n",
    "    num_solutions += 1\n",
    "    print('total_sum:', total_sum.Value())\n",
    "    print('row_sums:', [row_sums[i].Value() for i in range(rows)])\n",
    "    print('col_sums:', [col_sums[j].Value() for j in range(cols)])\n",
    "    print('row_signs:', [row_signs[i].Value() for i in range(rows)])\n",
    "    print('col_signs:', [col_signs[j].Value() for j in range(cols)])\n",
    "    print('x:')\n",
    "    for i in range(rows):\n",
    "      for j in range(cols):\n",
    "        print('%3i' % x[i, j].Value(), end=' ')\n",
    "      print()\n",
    "    print()\n",
    "\n",
    "  solver.EndSearch()\n",
    "\n",
    "  print()\n",
    "  print('num_solutions:', num_solutions)\n",
    "  print('failures:', solver.Failures())\n",
    "  print('branches:', solver.Branches())\n",
    "  print('WallTime:', solver.WallTime(), 'ms')\n",
    "\n",
    "\n",
    "main()\n",
    "\n"
   ]
  }
 ],
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   "name": "python"
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